178 CHAPTER 16
study group representations.
Today, his name is used for (among
other things) particular elements in the absolute Galois group G,
and also their restriction to the Galois groups of various polyno-
mials, that is, for particular permutations of the roots of various
Whenever you can get something for nothing in mathematics,
you take it. Of course, “nothing” is a relative term. What we
mean in this case is that we can deﬁne certain elements of every
Galois group by using an easily applied general theorem that tells
us signiﬁcant information about them. These are the Frobenius
To take a simpler example ﬁrst, let L be the ﬁeld Q(f )forsome
Z-polynomial f . We know that L is composed of certain complex
numbers. Let τ be complex conjugation.
Then τ deﬁnes an element
of the Galois group G(f )ofL:Ifx + iy is any number in L, τ (x + iy)
is again a number in L,andτ respects addition, subtraction,
multiplication, and division. Also, we have a formula for it: τ (x +
iy) = x − iy. Thus, τ is always there, and it is an important element
of the Galois group G(f ). It is the neutral element if and only if
L actually contains only real numbers, because then L does not
contain any numbers of the form x + iy with y = 0. W e can make τ
into an honorary Frobenius element at inﬁnity, and call it Frob
This is purely a notational convention, and does not help us to
when p is a prime. However, this notation Frob
common in the research literature.
The other Frobenius elements are at p for each prime p,and
we now begin to deﬁne them. If you choose to skip the rest of this
discussion, you need to know only:
is a particular element of G.
Actually, this is a lie, because we cannot deﬁne Frob
precisely. Really, given p, there is deﬁned a certain
conjugacy class in G (depending on p), and Frob
In fact, Frobenius deserves credit for inventing group representations themselves.
We have used the letter τ before to stand for various things, but here we will use
it for the speciﬁc function of complex conjugation. We have been using c for complex
conjugation, but from now on we prefer Greek letters for elements of Galois groups.